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Analogousexpressionsforturbulentheatandsalinityfluxare
w
T
0
=
α
h
u
∗
0
δ
T
(6.4)
w
S
0
=
α
S
u
∗
0
δ
S
where
S
0
arethedifferencesintemperatureandsalin-
ity between the far-field and interface values. If we assume that the analog holds
exactlyso thatscalar andmomentumexchangecoefficientsare comparable(a vari-
ant on Reynolds analogy), then it becomes clear why early ice/ocean models (e.g.,
Josberger 1983; Ikeda 1986) assumed that mixed-layer temperature remained near
freezing. Suppose that
δ
T
=
T
w
−
T
0
and
δ
S
=
S
w
−
01ms
−
1
,and
1K.During
the 1984 MIZEX experiment in the marginal ice zone of the Greenland Sea, sim-
ilar conditions persisted for at least a day as our floe drifted south over an ocean
front(seeFig.5.8andMcPheeetal.1987).With
S
ice
=
α
h
≈
α
m
=
0
.
11
,
u
∗
0
=
0
.
δ
T
=
0,thiswould
imply a melt rate of about 1.3m per day; however, we observed only about 7cm
of bottom ablation (corroborated by direct turbulent heat flux measurements). It is
worth recountingthat prior to MIZEX we had consideredthat the exchangecoeffi-
cients, couched in terms of reduced roughnesslengths for scalar variablesby anal-
ogy with atmosphericstudies, would be much smaller than for momentum(Mellor
etal. 1986);nevertheless,whenwe driftedoverwarm water duringthe later partof
MIZEX,the observedmeltrate and oceanheat fluxwere muchless than we antici-
pated. In retrospect, the fact that our floe survivedwhen theory suggested it should
have melted began the quest for a proper description of the exchange coefficients
forheatandsalt thatstill continues.
If the exchange coefficients for scalar quantities are different from momentum,
dimensional analysis (e.g., Barenblatt 1996) may help in suggesting the functional
form.Supposewechooseasourscalarquantitythekinematicheatfluxandpostulate
thatit dependsonthefollowingquantities:
w
T
0
=
3psu,and
q
=
(
δ
,
u
∗
0
,
ν
,
ν
,
)
F
T
z
0
T
ThenbythePiTheorem,therearefivegoverningparameters,threewithindependent
dimensions,sothat
T
=
α
h
ν
w
T
0
u
∗
0
δ
u
∗
0
z
0
ν
ν
T
,
=
α
h
(
Pr
,
Re
∗
)
(6.5)
where
Pr
and
Re
∗
are the Prandtl and surface friction Reynolds numbers, respec-
tively.Asimilar expressionforsalinityfluxis
S
ν
w
S
0
u
∗
0
δ
u
∗
0
z
0
ν
S
=
α
ν
S
,
=
α
(
Sc
,
Re
∗
)
(6.6)
S
where
Sc
istheSchmidtnumber.ThePrandtlandSchmidtnumberdependencysug-
geststhatforscalarquantities,moleculardiffusivitiesmaybeimportantandindeed
thisturnsouttobethecase.Notethat
α
S
representtheinverseofdimension-
lesschangesin temperatureandsalinityacrosstheboundarylayer.
α
h
and
